Signatures of Chemical Dopants in Simulated Resonance Raman Spectroscopy of Carbon Nanotubes

Single-walled carbon nanotubes (SWCNTs) with organic sp2 or sp3 hybridization defects allow the robust tunability of many optoelectronic properties in these topologically interesting quasi-one-dimensional materials. Recent resonant Raman experiments have illuminated new features in the intermediate-frequency region upon functionalization that change with the degree of functionalization as well as with interactions between defect sites. In this Letter, we report ab initio simulated near-resonant Raman spectroscopy results for pristine and chemically functionalized SWCNT models and find new features concomitant with experimental observations. We are able to assign the character of these features by varying the frequency of the external Raman laser frequency near the defect-induced E11* optical transition using a perturbative treatment of the electronic structure of the system. The obtained insights establish relationships between the nanotube atomistic structure and Raman spectra facilitating further exploration of SWCNTs with tunable optical properties tuned by chemical functionalization.

: Normalized off-resonant Raman spectra for pristine and defected SWCNT systems. Normalization condition is defined as I → I / MAX(I). Figure S2: Normalized pre-resonant Raman spectra using the same as presented in Fig. 4. Normalization is defined as I → I / MAX(I). In experiment, usually, the normalization is usually defined as I → I / I("G-Mode"), but for our spectra, the definition of the G-Mode is not welldefined, especially for perturbation energies larger than 1.20 eV. Figure S3: Raman intensity for a variety of major peaks that exhibit strong frequency dependence for the (a) pristine and (b) defected SWCNT models. The data is the same as that shown in Figure  4 in the Main Text with each curve normalized to its own maximum. The labels of the curves correspond to the labels shown in Figure 4.    One can compute the different between the pristine and defected SWCNT amplitude density maps, which is presented in Fig. S8. Here, the defect modes are clearly visualized as a function of the normal mode frequency and SWCNT axis position with most effects stemming from the edges present in both the pristine and defected SWCNTs can be eliminated. Although, some edge effects are still seen, possibly due to the choice of discretization grid and can roughly be ignored since the adjacent positive and negative regions would exactly cancel for the peaks at the edges if binned together. The current spatial resolution is useful for examining the central portion of the SWCNT. Figure S8: Pre-resonance Raman spectra for the (a) pristine and (b) defected SWCNT models replotted from Fig. 4 in the main text. The colors represent the CPHF perturbation frequency. The open black circles indicate the normal mode extent Ld parameter as a percent of the total SWCNT length. The mode near 900-950 cm -1 and 1800 cm -1 exhibit the most localized modes, having only ~10% or less extent along the SWCNT. The majority of features exhibit more than 40 % delocalization. Figure S9: Real-space projected ground-to-excited transition density for the lowest five excitonic transitions. The transition energy and oscillator strength (in parenthesis) are shown for each transition. The S1 is labeled as the E11 * transition, and the S5 is labeled as the E11 transition. The isosurface value is 0.0008.  Figure S11: Intensity difference defined from the pre-resonant Raman spectra for the near E11 and E22 perturbation energies.
In the presence of an oscillating external field ( , ω), one can perform a perturbative expansion around the electric field amplitude for the energy of the system, where ϵ 0 is the energy of the system without the presence of an external electric field, is a component of electric field vector, μ , α , β , γ are the tensor components of the dipole, polarizability, first hyperpolarizability, and second hyperpolarizability of the molecular system. 12 These quantities can be interpreted as molecular response functions dependent on the electric field . The total energy, as well as all future quantities, is assumed to be a sum of all components, abcd, following Einstein sum rules unless otherwise stated. Similarly, the total molecular polarization ∼ − ∂ϵ( )/ ∂ can be written as, The polarization ( ) can be extracted from a wavefunction as, where Here, _ is the total Hamiltonian of the system in the absence of the external field, and ′ = − ⋅ . From here, one can defined the ground state density matrix in terms of the single slater determinant (for Hartree-Fock-or Kohn-Sham-like calculations) as, with matrix elements defined as ρ = * , { } molecular orbital coefficients, and = 2δ for j < NOcc. and = 0 for j > NOcc. as the occupation matrix. Now, the molecular polarization can be written as, where μν = − ⟨μ|̂|ν⟩ is the dipole matrix in the atomic orbital basis {μ}. Finally, the density matrix itself can be decomposed into contributions from each order of interaction with the electric S13 and field and all necessary quantities can then be calculated from this decomposition. The density matrix is, and inserting into Eq. 5, we obtain the tensor polarizabilities as, where the ω frequencies can be either the static frequeny (i.e., 0) or the forward or backward external electric frequencies (i.e., −ω, ω). Higher-order frequencies that are integer multiples of the external frequency can be calculated using rules from generic perturbation theory to simplify the problem. 14 The main challenge is obtaining the n th -order density matrix ρ and is the main inspiration for solving the coupled perturbed Hartree-Fock (CPHF) equations. The explicit details of these equations can be found in ref. 13 . Here, only the main equations in a truncated form will be presented in the language of HF but can be easily adapted for DFT. The Fock matrix equation subject to a time-dependent interaction can be written as, where C are the time-dependent molecular orbital expansion coefficients, F is the Fock matrix, and S is the overlap matrix of the basis functions. The molecular orbitals obey the following orthogonalization condition: After expanding in orders of electric field interaction, as above, we obtain the CPHF equations for the Fock matrix can be written as,
Once the perturbed density matrix is calculated, the tensor polarizabilities can be obtained through various orders of differentiation of the density matrix with respect to the electric field. In principle, these four tensors allow one to calculate a variety of non-linear spectra. In the present work, we limit ourselves to the calculation of Raman spectroscopy, which will only make use of the polarizability α . To make a simple connection to Raman spectroscopy, the state-to-state polarizability can be calculated in the language of normal modes { } as, Where | ⟩ and | ⟩ are the initial and final states, respectively. This expressions indicates that the Raman intensity will only be non-zero for a n-to-m transition if there exists a change in the polarizability with respect to nuclear configuration as well as having an appreciable dipole moment between n and m states. This quantity can often be obtained from analytical gradient expressions with respect to nuclear coordinates implemented in many software packages for HF and DFT levels of theory.

Normal Mode Character
The normal modes of each model (pristine and defected) were extracted with the vectorial expansion coefficients for each atom written as, where α is a nucleus and is the normal mode index. The norm of each atom contribution is then computed as, A discrete probability distribution ( ) = ( ) 2 can be constructed by binning the contributions from all atoms within a spatial histogram (X b ) at bin X b of width = 0.45 Å (100 bins spanning the length of the SWCNT main axis) for each mode defined as, S15 for all α in bin . The normalization of the probability distribution ( ) for each mode is defined as, This probability distribution will be referred to as a position-resolved normal mode probability distribution and will be able to give insight into the spatial localization of each normal mode.
where σ = 5 cm -1 . This allows to visualize the spatial distribution of the modes as a function of the normal mode frequency. Fig. S7 shows the same data as a 3D contour plot for visual comparison to the 2D density map in Fig. 5 in the main text for the defected SWCNT.
To gain a better understanding of which types of mode characters are contributing to the intensity of the CPHF spectrum, we further decompose the position-resolved probability distribution into three classes of modes: (I) Edge-localized, (II) Defect-(center-)localized, and (III) all others.
If the normal mode is localized to within 5 Å of the edge by more than 35 % (sum of the probabilities of both edges), the intensity of this mode for each value of CPHF energy is shown as the percent of the maximum intensity in the system in Fig. 5 for the (a) pristine and (b) defected SWCNT models. If the normal mode is localized to the center 10 Å (5 Å on either side of the center of the SWCNT) by more than 30 %, the intensity of this mode for each value of CPHF energy is shown as the percent of the maximum intensity in the system in Fig. 5 for the (c) pristine and (d) defected SWCNT models.
One can define a localization parameter, similar to that of the inverse participation ratio commonly used in the description of quantum delocalization. [1][2][3][4] We define the Ld parameter for the th normal mode as, which can be converted to length by ( ) → ( ) * or to a percent of the total SWCNT axis as ( ) → ( ) * , where is the length of the SWCNT axis. Here, the larger the percent ( ) (%) the more delocalized the mode along the SWCNT axis; whereas, a small percent indicates that the mode is localized to only that percent of the SWCNT. This data is shown in Fig.  S9.